Celebratio Mathematica

U. Grenander and M. Rosenblatt: “Some problems in estimating the spectrum of a time series,” pp. 77–93 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 1: Contributions to the theory of statistics. Edited by J. Nyman. University of California Press (Berkeley and Los Angeles, CA), 1956. MR 84914 Zbl 0072.36401 incollection

Jerzy Neyman	Related
Ulf Grenander	Related

M. Rosenblatt: “Some regression problems in time series analysis,” pp. 165–186 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 1: Contributions to the theory of statistics. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles, CA), 1956. MR 84920 Zbl 0071.35602 incollection

Estimates of the regression coefficients which are unbiased and linear in the observations are discussed in this paper. The residual is assumed to be a stationary process. Two specific estimates are discussed, the least-squares estimate and the Markov estimate. I call the estimate which is computed under the assumption that the residual is an orthogonal process the least-squares estimate. The Markov estimate is the linear unbiased estimate with minimal covariance matrix. The basic assumptions made in the paper are discussed in section 2 and are held to throughout the paper. In section 3 some remarks about the approximation of a continuous positive definite matrix-valued function by finite trigonometric forms are made. These remarks are used in section 4 to obtain the main results about the asymptotic behavior of the covariance matrices of the least-squares and Markov estimates. The next section discusses the many interesting cases in which the least-squares estimate is asymptotically as good as the Markov estimate. The first really systematic discussion of some of these problems was given by U. Grenander [1954]. Further work was carried out by U. Grenander and M. Rosenblatt in [1954a, 1954b, 1957]. The author considers some of these problems in the case of a vector-valued time series in [1956]. Some of the results of this paper are a generalization of some of those obtained in [1956].

A few cases in which the least-squares estimate is not asymptotically efficient in the class of linear unbiased estimates are discussed in sections 5 and 7. Some small sample computations for a linear regression with a residual which is a first order autoregressive scheme are carried out in section 6 to test the asymptotic theory.

M. Rosenblatt: “Independence and dependence,” pp. 431–443 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 20–30 July 1960), vol. 2. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 133863 Zbl 0105.11802 incollection

A stochastic process is commonily used as a model in studying the behavior of a random system through time. It will be convenient for us to take the stochastic process \( \{x_t\} \) as discrete in time \( t = \dots, -1 \), 0, \( 1,\dots \). Processes of independent random variables are the simplest and most completely understood. It is, however, clear that these are extremely limited in scope as models and one must have recourse to dependent processes (the random variables \( x_t \), not independent) in order to have any power in description. For simplicity, let us further restrict ourselves to processes that are stable through time, stationary processes. For such processes the probabilities of events shifted through time remain the same, that is, the probability \[ P\{x_{t_1+h} \leq a_1,\dots,x_{t_n+h} \leq a_n\} \] is independent of \( h \). Such models occur fairly often in the physical sciences. If mean properties of the process are to be capable of being estimated reasonably well from part of a realization of the process, some form of asymptotic independence for blocks of random variables of the process that are widely separated must be satisfied. This is, in effect, the gist of many of the results in ergodic theory. Two types of interesting problems are posed.

The first of these is concerned with reasonable notions of asymptotic independence and what types of processes satisfy them.

The second is that of characterizing those processes \( \{x_t\} \) that can be constructed out of independent processes by a function and its shifts, that is, \[ x_t = (\dots, \xi_{t-1},\xi_t,\xi_{t+1},\dots) \] where \( \{\xi_t\} \) is a process of independent random variables. Neither of these questions have elicited satisfactory answers. However, there are some small results that do give insights into the problems. The object of this paper is a presentation and discussion of a few of these limited results.

M. Rosenblatt: “Transition probability operators,” pp. 473–483 in Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1965 and 27 December 1965–7 January 1966), vol. 2: Contributions to probability theory, part 2. Edited by L. M. Le Cam and J. Neyman. University of California Press (Berkeley, CA), 1967. MR 212877 Zbl 0222.60042 incollection

Lucien Marie Le Cam	Related
Jerzy Neyman	Related

M. Rosenblatt: “Central limit theorem for stationary processes,” pp. 551–561 in Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1970), vol. 2: Probability theory. Edited by L. M. Le Cam, J. Neyman, and E. L. Scott. University of California (Berkeley), 1972. MR 402869 Zbl 0255.60026 incollection

Elizabeth Leonard Scott	Related
Lucien Marie Le Cam	Related
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